Finite Fields

This chapter describes the special functionality which exists in &GAP; for finite fields and their elements. Of course the general functionality for fields (see Chapter ) also applies to finite fields.

In the following, the term finite field element is used to denote &GAP; objects in the category , and finite field means a field consisting of such elements. Note that in principle we must distinguish these fields from (abstract) finite fields. For example, the image of the embedding of a finite field into a field of rational functions in the same characteristic is of course a finite field but its elements are not in , and in fact &GAP; does currently not support such fields.

Special representations exist for row vectors and matrices over small finite fields (see sections and ).

Finite Field Elements
<#Include Label="IsFFE"> <#Include Label="Z"> <#Include Label="IsLexOrderedFFE">

Operations for Finite Field Elements
<#Include Label="[2]{ffe}"> <#Include Label="DegreeFFE"> <#Include Label="LogFFE"> <#Include Label="IntFFE"> <#Include Label="IntFFESymm"> <#Include Label="IntVecFFE"> <#Include Label="AsInternalFFE">

Creating Finite Fields
<#Include Label="DefaultField:ffe"> <#Include Label="GaloisField"> <#Include Label="PrimitiveRoot">

Frobenius Automorphisms
<#Include Label="FrobeniusAutomorphism">

Conway Polynomials
<#Include Label="ConwayPolynomial"> <#Include Label="IsCheapConwayPolynomial"> <#Include Label="RandomPrimitivePolynomial">

Printing, Viewing and Displaying Finite Field Elements
Internal finite field elements are viewed, printed and displayed (see section for the distinctions between these operations) as powers of the primitive root (except for the zero element, which is displayed as 0 times the primitive root). Thus:

gap> Z(2); Z(2)^0 gap> Z(5)+Z(5); Z(5)^2 gap> Z(256); Z(2^8) gap> Zero(Z(125)); 0*Z(5)

Note also that each element is displayed as an element of the field it generates, and that the size of the field is printed as a power of the characteristic.

Elements of larger fields are printed as &GAP; expressions which represent them as sums of low powers of the primitive root:

gap> Print( Z(3,20)^100, "\n" ); 2*Z(3,20)^2+Z(3,20)^4+Z(3,20)^6+Z(3,20)^7+2*Z(3,20)^9+2*Z(3,20)^10+2*Z\ (3,20)^12+2*Z(3,20)^15+2*Z(3,20)^17+Z(3,20)^18+Z(3,20)^19 gap> Print( Z(3,20)^((3^20-1)/(3^10-1)), "\n" ); Z(3,20)^3+2*Z(3,20)^4+2*Z(3,20)^7+Z(3,20)^8+2*Z(3,20)^10+Z(3,20)^11+2*\ Z(3,20)^12+Z(3,20)^13+Z(3,20)^14+Z(3,20)^15+Z(3,20)^17+Z(3,20)^18+2*Z(\ 3,20)^19 gap> Z(3,20)^((3^20-1)/(3^10-1)) = Z(3,10); true

Note from the second example above, that these elements are not always written over the smallest possible field before being output.

The and methods for these large finite field elements use a slightly more compact, but mathematically equivalent representation. The primitive root is represented by z; its i-th power by z i and k times this power by k z i.

gap> Z(5,20)^100; z2+z4+4z5+2z6+z8+3z9+4z10+3z12+z13+2z14+4z16+3z17+2z18+2z19

This output format is always used for . For it is used only if its length would not exceed ViewLength() lines. Longer output is replaced by <<an element of GF(p, d)>>.

gap> Z(2,409)^100000; <<an element of GF(2, 409)>> gap> Display(Z(2,409)^100000); z2+z3+z4+z5+z6+z7+z8+z10+z11+z13+z17+z19+z20+z29+z32+z34+z35+z37+z40+z\ 45+z46+z48+z50+z52+z54+z55+z58+z59+z60+z66+z67+z68+z70+z74+z79+z80+z81\ +z82+z83+z86+z91+z93+z94+z95+z96+z98+z99+z100+z101+z102+z104+z106+z109\ +z110+z112+z114+z115+z118+z119+z123+z126+z127+z135+z138+z140+z142+z143\ +z146+z147+z154+z159+z161+z162+z168+z170+z171+z173+z174+z181+z182+z183\ +z186+z188+z189+z192+z193+z194+z195+z196+z199+z202+z204+z205+z207+z208\ +z209+z211+z212+z213+z214+z215+z216+z218+z219+z220+z222+z223+z229+z232\ +z235+z236+z237+z238+z240+z243+z244+z248+z250+z251+z256+z258+z262+z263\ +z268+z270+z271+z272+z274+z276+z282+z286+z288+z289+z294+z295+z299+z300\ +z301+z302+z303+z304+z305+z306+z307+z308+z309+z310+z312+z314+z315+z316\ +z320+z321+z322+z324+z325+z326+z327+z330+z332+z335+z337+z338+z341+z344\ +z348+z350+z352+z353+z356+z357+z358+z360+z362+z364+z366+z368+z372+z373\ +z374+z375+z378+z379+z380+z381+z383+z384+z386+z387+z390+z395+z401+z402\ +z406+z408

Finally note that elements of large prime fields are stored and displayed as residue class objects. So

gap> Z(65537); ZmodpZObj( 3, 65537 )